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Unformatted text preview: WAVE FUNCTIONS AND PROBABILITY 1 I: Thinking about the wave function In quantum mechanics, the term wave function usually refers to a solution to the Schr odinger equation, i ~ ( x,t ) t =- ~ 2 2 m 2 ( x,t ) x 2 + V ( x )( x,t ) , where V ( x ) is the potential energy experienced by a particle of mass m and ( x,t ) is the wave function in this one-dimensional example. A. Lets say you have a system where the wave function is of the form: 1 ( x,t ) = f ( x ) e it where f ( x ) is some real-valued function of x . 1. Is | 1 ( x,t ) | 2 real? Is it positive? Do your answers make sense given the physical meaning (as discussed in class) of | 1 ( x,t ) | 2 ? 2. Does 1 ( x,t ) depend on time? Does | 1 ( x,t ) | 2 depend on time? 3. Write down an expression for h x i . Does it depend on time? Is it real? Describe in words how you interpret this quantity. Precisely what information do you get from h x i ? 4. Write down an expression for h g ( x ) i where g ( x ) is any real-valued function of x . Does it depend on time? Again, how would you physically interpret h g ( x ) i (hint: think about what you would actually measure)? PHYS 3220 Tutorials 2008 - 2010 c S. Goldhaber, S. Pollock, and the Physics Education Group University of Colorado, Boulder WAVE FUNCTIONS AND PROBABILITY 2 B. Now lets say your system is a bit more complex (pun intended): 2 ( x,t ) = f ( x ) e it + g ( x ) e 2 it...
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